Binding energies in nonrelativistic field theories

ELSEVIER

Nuclear Physics B (Proc. Suppl.) 53 (1997) 401-404

SU

MENTS

Binding Energies in Nonrelativistic Field Theories Andreas S. Kronfeld a a Theoretical Physics Group, Fermi National Accelerator Laboratory, Batavia, Illinois, USA Relativistic corrections communicate the binding energy of a bound state to its kinetic mass. This mechanism is reviewed and used to explain anomalous results of Collins, Edwards, Heller, and Sloan (hep-lat/9512026), which compared rest and kinetic masses of heavy-light mesons and quarkonia.

i. INTRODUCTION Last year Collins, Edwards, Heller, and Sloan [1] studied heavy Wilson quarks [2] with the improved action of Sheikholeslami and Wohlert [3]. In lattice units the heavy-quark mass mQa typically exceeded unity, a regime in which the numerical results require a nonrelativistic interpretation [4,5] just as in NRQCD [6,7]. Ref. [1] presents a test of the nonrelativistic interpretation that removes kinematic effects and focuses on a dynamical effect--the binding energy. The results of this test were unexpectedly anomalous. T h e aim of this paper is to explain why and to offer a remedy. Let us begin w i t h some basics to define notation. As a function of m o m e n t u m p the energy of a state X can be written

E x ( p ) = M i x + 2M2---~

8---~4zx+ . . . ,

(1)

where the rest mass is defined by M1 = E(0), and the kinetic mass is defined by

(02E

-1

M2 = \ Op~ Jp=o"

(2)

Below the states can be quarks, Q and q, and mesons ~)Q, Oq, and qq. Usually Q is assumed heavier than q. On a relativistic mass shell

M i x = M2x = "" = reX,

(3)

In this paper the lower-case m x denotes the exact, physical mass, whereas upper-case M i x denote the result of a (possibly approximate) calculation. In the mass-dependent renormalization 0920-5632(97)/$17.00 © 1997 Elsevier Science B.~ All rights reserved. PIl: S0920-5632(96)00671-8

of ref. [5] it is possible to adjust the action's couplings to recover eq. (3), to a specified accuracy. With the Wilson or Sheikholeslami-Wohlert actions, however, perturbation theory shows that MIO ~ M2Q (except as m o a --~ 0). In nonrelativistic systems this is acceptable, provided one adjusts the bare mass until the kinetic mass takes its physical value [4,5], just as in ref. [6,7]. The quark state makes sense at most in perturbation theory. In a nonperturbative world, one would like to carry out the tuning with a bound state, e.g. a meson X = 0 q whose masses Ml(~q , M2¢q, etc, can be computed with Monte Carlo methods. Let us define the binding energy B by

MlCq = M I ¢ + Mlq + BI, M2(~q = M2(~ + M2q + B2.

(4)

To make a precise definition of the binding energies, one requires a precise definition of the quark masses in eq. (4). Here it is enough to take the rest and kinetic masses of the free theory (g2 = 0) with the same bare quark masses. Computing bound-state splittings through rest masses makes sense only if B1 is the experimental binding energy, to sufficient accuracy. Similarly, tuning the meson mass to the kinetic mass makes sense only if also B2 is the experimental binding energy, to sufficient accuracy. As shown below, the anomalous result of ref. [1] is sensitive to B2 - B1. There are two keys to understanding it. First, one must be careful about the qualifying phrase "to sufficient accuracy" in the preceding paragraph. Second, one must to know how field theories communicate the binding energy to a bound state's kinetic mass.

A.S. Kronfeld/NuclearPhysicsB (Proc. Suppl.) 53 (1997)401-404

402

Sect. 2 recalls the diagnostic test of ref. [1]. Sect. 3 assesses the cutoff effects of the binding energy in light-light and heavy-light mesons, and in quarkonium. The mechanism for generating the "kinetic binding energy" is reviewed for a relativistic (continuum) gauge theory in sect. 4 and generalized to lattice gauge theory in sect. 5. Sect. 6 draws a few conclusions.

0.2

....

'''l

. . . .

I . . . .

I

,,,I

....

I ....

I ....

2

3

0 -0.2 -0.4

2. T H E

TEST

-0.6

Let us abbreviate 5M :=/1//2 - M1 and 5B := B2 - B 1 . Ref. [1] introduces

z := 2 Moq -

+

2M~Qq

-0.8

(5)

Comparison with eqs. (4) shows that the quark masses drop out, leaving

I = 25BQq - (6BQQ + 5Bqq) 2M20q

(6)

If the lattice action(s) of the quarks were sufficiently accurate, all 5Bs, and hence I, would vanish. (I vanishes trivially when mQ = mq, even if

BQq # 0.) The numerical results of ref. [1] are shown in fig. 1. The "inconsistency" I is negative, and [I[ tends to increase with increasing mQ. To explain both the sign and the magnitude, below I shall derive an expression for 5B. 3. C U T O F F E F F E C T S O N 5B Before presenting the analytical result for 5B, it is useful to anticipate the order of magnitude of 5B in each meson--light-light, heavy-light, and quarkonium. On this basis it turns out that the quarkonium 5B~Q dominates the numerator in

eq (6) Light-light 5Bqq The binding energy is O(AQCD). With the Sheikholeslami-Wohlert action, B1 and B2 both suffer from lattice artifacts of O(c~naAQCD). With the tree-level improvement of used by ref. [1], n = 1. (With the Wilson action n = 0.) There is no reason for the artifacts to be identicai, so 6Bqq is O(anaA~cD). This is numerically small, so 6Bqq can be neglected below. 3.1.

0

1

4

aM2 Qq Figure 1. Plot of the binding-energy "inconsistency" I vs. the heavy-light meson's kinetic mass ai2~2q , for mQ >>mq. Adapted from ref. [1]. 3.2.

Heavy-light

5BQq

The binding energy is again O(AQcD). The light quark suffers lattice artifacts as above, but, when mQ ~ 1--as in fig. 1, the heavy quark also suffers from (smaller) effects of O(A~cDa/mQ). Again, even though there is no reason for artifacts in B1 and B: to cancel, one sees that 5B~q is numerically negligible. 3.3. Q u a r k o n i u m 5BQQ The binding energy is now O(mQv2), where v denotes the relative Q-Q velocity. In this nonrelativistic system, the velocity is a pertinent estimator of cutoff effects [7,5]. The rest mass is O(v°), so the action would need absolute accuracy of O(v 2) to obtain relative accuracy of O(v 2) in BI. Both the Wilson and Sheikholeslami-Wohlert actions achieve this. On the other hand, the kinetic mass multiplies an O(v 2) effect, so the action would now need an absolute accuracy of O(v 4) to obtain relative accuracy of O(v 2) in B2. Neither the Wilson nor the Sheikholeslami-Wohlert action achieves this [5]; with either of them, one can only hope for O(v °) relative accuracy in/32. The error in B2, and hence in 5B£2Q =/=O, is O(mQv2), which is significantly larger than the previous two estimates.

403

A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 53 (1997) 401-404

4. B R E I T E Q U A T I O N For nonrelativistic systems the binding-energy discrepancy can be worked out quantitatively, following a textbook nonrelativistic expansion of QED [8]. This section verifies in a relativistically invariant theory that B2 = B1 = B. The next section then turns to the lattice theories, which break relativistic invariance. For convenience, these two section assume that even the "light" quark q is nonrelativistic. At leading order the quark-anti-quark interaction arises from one-gluon exchange diagrams. Evaluating these diagrams and developing the nonrelativistic expansion, one obtains a Hamiltonian H = m R + mq + H2 + H4 for the quarkanti-quark system. The leading nonrelativistic dynamics are given by H2 = 2 - -

+

2rnq p2

-- 2UQq "b ~

+ V(r) (7) -}" V ( r ) ,

where V(r) = --CFOt/r; r

=

~rQ -- Xq, P and p

are center-of mass coordinates and momentum; p = (m~ 1 + mql) -1 is the reduced mass; and MQq = mo + m q. The first relativistic corrections are

H4=

8m3

8rn~ + V2(r;pQ,pq),

(S)

where the non-local potential V2 is given by Breit's equation [8]. It takes the form

L

+

r-2rirjpQipq j

p2

(9)

2MQqp

+ spin-dependent. The spin-dependent terms and the terms proportional to J(3)(r) are not important here. Full details are given in §§ 83-84 of ref. [8]. Together with the (p2)2 terms in /'/4, the exhibited part of V2 is responsible for modifying the bound-state kinetic mass from MOq = m R + mq to m(~q = m~2 + mq + B (as required by Lorentz invariance).

p2

(


2m(~q "-- 2U(~q 1

Uoq ]

(10)

PiPj ( r i V j Y - p i p j / p } where T = p2/2p is the internal kinetic energy. By the virial theorem the second line vanishes. Thus, to consistent order in p / M the leading relativistic corrections/-/4 generates the right binding energy B2 = {T + V} =: B for the bound-state kinetic mass. More generally, higher-order relativistic effects trickle down to bound-state properties as follows: the correction of O(v e) provides the O(v e-k) contribution to bound-state properties of O(vk). 5. L A T T I C E G E N E R A L I Z A T I O N On a hypercubic lattice there can be two corrections to the kinetic energy

E(V) =

(p2)2 8M~

, 3 gwaa Z p ~ + . ' . ,

(11)

i

for each of p = PQ' Pq" Here 04 E ) - , / 3 M4 :=

V2(r;pQ,pq)= const X 5(3)(r) + V(r) [1 - po "pq

To proceed one must re-write H4 in center-ofmass momenta and collect terms quadratic in the total bound-state momentum P . In the bound state, combinations of the internal momentum p and relative coordinate r can be replaced by expectation values. Collecting all terms, the boundstate kinetic energy becomes

--

--'~-2

OPiOPJ p=O

,

i# j

(12)

and

1 04E[ w4 :=

4 0 p ~ p=,

3 4 M 2"

(13)

Unless the action has been improved further than the Sheikholeslami-Wohlert action, M4 ~ M2 and w4 5~ 0, el. Appendix A of ref. [5]. These lattice artifacts filter through to B2--just as above-through the terms proportional to PIPi(PiPj>. The spatial gluon generates the spinindependent contribution proportional to V(r) in

404

A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 53 (1997) 401-404

eq. (9); on the lattice the nuance is that the kinetic mass appears in the bracket. On the other hand, the temporal gluon generates more complicated terms, but they either depend on spin or are proportional to 8 (3) (r). So to work out an expression for B2, it is enough to maintain eq. (9), but with masses M2~q and p~ built from M2~ and M2q. The calculation of the binding energy difference 5B follows the steps leading to eq. (10). One finds an expression that is too cumbersome to present here. In an S wave, however, it can be simplified because (PiPj) = ~Sij(p2) • Then

(T) = ~

{

5

2 k Ma4Q + M~q] - I

] %

+

5BQQ ~, -0.5. 2Moq

ACKNOWLEDGEMENTS

I would like to thank Sara Collins and John Sloan for bringing their results to my attention and for insisting that I explain it. Fermilab is operated by Universities Research Association, Inc., for the U.S. Department of Energy.

(14) REFERENCES

+ M qw,q)I"

This is the main new result of this paper. Note that, as one would have anticipated, the expression vanishes when w 4 x = 0 and M 4 x = M2x. With an estimate of (T) from potential models [9] and the lattice masses of the right-most point in fig. 1, 1 find

I ~

(or higher). In particular, one requires M4 = M2 and w4 = O. Most published applications of ref. [6] use a sufficiently accurate action [7]. Ref. [10] even remarks that O(v 4) accuracy is essential for a consistent determination of the b-quark mass from the T spectrum. For four-component fermions the details required for O(v a) accuracy in quarkonium have appeared more recently [5].

(15)

The agreement with the Monte Carlo results is surprisingly good. 6. C O N C L U S I O N S The origin of the anomaly observed in ref. [1] is the usage of an action accurate only to O(v2). Thus the relative error in the binding energy B2 of the bound-state kinetic mass is of order mQv2/rnQv 2 = 1. Meanwhile, the usual binding energy B1 is indeed valid to leading order in v 2. The test quantity I cleverly isolates B~ - B 1 , and thus exposes an inconsistency of O(1). By examining how (approximately) relativistic field theories generate B~, this paper explains the results found last year [1]. Moreover, the analysis makes the remedy plain: the anomaly is not expected to appear if quarkonium properties are computed with an action improved through O(v 4)

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